nonlinear dynamics of wrinkle growth and pattern formation
TRANSCRIPT
Nonlinear Dynamics of Wrinkle Growth and Pattern Formation in Stressed Elastic Thin FilmsPattern Formation in Stressed Elastic Thin Films
Se Hyuk Im and Rui Huang
Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics
Th U i it f T t A tiThe University of Texas at Austin
----- Work supported by NSF CAREER Award (CMS-0547409) -----
Au film on PDMS: thermoelastic wrinkling• Bowden et al., 1998 & 1999• Huck et al., 1999• Efimenko et al., 2005
SiGe film on BPSG: viscous wrinkling• Hobart et al 2000Hobart et al., 2000• Yin et al., 2002 & 2003• Peterson et al., 2006• Yu et al., 2005 & 2006
BPSGSi
SiGe
Si
Heat to 750°C
Elastic wrinkling vs Viscous wrinkling
film/substrate materials elastic/elastic elastic/viscous
Onset of wrinkling Critical stress or strain Unstable with any gdepends on the elasticmodulus and thickness of substrate.
ycompressive stress in a thin film (L/h >> 1).
Wrinkle amplitude Equilibrium amplitude increases with the stress/strain.
Amplitude grows over time.
W i kl l th E ilib i l th Th f t t iWrinkle wavelength Equilibrium wavelength depends on the elasticmodulus and thickness of substrate.
The fastest growing wavelength dominates the initial growth, then coarsening over time.g
Energetics Spontaneous energy minimization.
Energy dissipation by viscous flow.
(Yoo and Lee, 2003-2005)
Al film on PS: viscoelastic wrinkling
A model for viscoelastic wrinklingViscoelastic relaxation modulusViscoelastic relaxation modulus Wrinkle
Amplitude Rubbery State
(II)Glassy State
Equilibrium states at the (I)
(II)
Compressive0
Rε Gε
Equilibrium states at the elastic limits (thick substrate):
3/23
41
⎟⎟⎠
⎞⎜⎜⎝
⎛−= s
c EEε
(I)
Compressive StrainEvolution of wrinkles:
(I) Viscous to Rubbery
R G
2/1
1⎟⎟⎠
⎞⎜⎜⎝
⎛−=
cfeq hAεε
4 ⎠⎝ fE
(I) Viscous to Rubbery
(II) Glassy to RubberyHuang, JMPS 2005.
⎠⎝3/1
32 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
s
ffeq E
Ehπλ
Scaling of evolution equationElastic film
( ) RwwFwKtw
−∇⋅⋅∇+∇∇−=∂∂ σ22
Viscoelastic layer
Elastic film
Rigid substratehh)21( 3
s
RRημ
=
Fastest growing mode: ⎟⎞
⎜⎛ t
Rigid substrate
sfs
sffs hhK
ηννμν
)1)(1(12)21( 3
−−
−=
s
fs
s
s hhF
ηνν
)1(221−−
=
Fastest growing mode:122 Lm πλ = ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
10 4exp
τtAA
K K
4KR
01 σF
KL −= ( )201
στ
FK
=Length and time scales:
24FKR
c −=σcritical stress:
K ⎞⎛4/1 ⎞⎛2Equilibrium state:
meq RK λπλ >⎟⎠⎞
⎜⎝⎛= 2 ⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 1
32 0
cfeq hA
σσEquilibrium state:
)( 0 cσσ <
Three-Stage Viscoleastic Wrinkle Evolution
Stage I Stage II Stage III
wavelengthwavelength
amplitude
Timet1 t2
I E ti l th d i t d b th f t t i dI: Exponential growth, dominated by the fastest growing mode
II: Coarsening (λm → λeq, A↑)
III: Nearly equilibrium (with local ordering)Im and Huang, JAM 2005 and PRE 2006.
Power-law Coarsening
1 σFKL −=
Length and time scaling:
0σF
( )201
στ
FK
=
σ0 = -0.02 (Δ), -0.01 (o), -0.005 (□), -0.002 (◊), -0.001 (∇)
Blue: uniaxial; Red: equi-biaxial
• On a compressible thin viscoelastic substrate: λ ~ t1/4;
q
On a compressible, thin viscoelastic substrate: λ t ;
• On an incompressible, thin viscoelastic substrate, λ ~ t1/6;
• On a thick VE substrate, λ ~ t1/3.,
• The substrate elasticity slows down coarsening as it approaches the equilibrium state.
Huang and Im, PRE 2006.
Numerical simulation: uniaxial wrinklingg
t = 104 t = 105t = 0 t = 104 t = 105t 0
t = 106 t = 107
0,01.0 )0(22
)0(11 =−= σσ
0=f
R
μμ
, 2211
fμ
Huang and Im, PRE 2006.
Numerical simulation: equi-biaxial wrinkling
t = 104 t = 105t = 0 t = 104 t = 105t 0
t = 106 t = 107
010)0()0( −==σσ
0=R
μμ
01.02211 ==σσ
fμ
Huang and Im, PRE 2006.
Effect of Substrate Elasticity 510−=f
R
μμ
σ0 = -0.02 (Δ), -0.01 (o), -0.005 (□)
Near-Equilibrium Patterns 510−=f
R
μμ
0
005.0)0(
22
)0(11
=
−=
σ
σ 005.0)0(22
)0(11 −== σσ
Same wavelength
Different patterns01.0)0(22
)0(11 −==σσ
02.0)0(22
)0(11 −==σσ
Huang and Im, PRE 2006.
Wrinkle patterns under biaxial stressesσy
nkle
sar
alle
l wrin
σc
Pa
0 σxσc
Parallel wrinkles
cy νσσ
=−Transition
σ2
fcx
νσσ
=−stress:
10 μm σ1
Wrinkle patterns under non-uniform stresses1D h l d l
0σσ =
)(xfx =σ
σ x/E
f( )( )⎥⎦
⎤⎢⎣
⎡−=
aLax
x 2/cosh/cosh10σσ
1D shear-lag model:
0σσ y
Stre
ss,
R
fsf hhEa
μ=
Lx/hf
005.0/0 −=fEσ0== yx σσoutside (green):
L = 3750 hf L = 3250 hf L = 2500 hf L = 1250 hf
Wrinkling of rectangular filmsE i t (Ch i t l 2007) Si l ti (I d H 2009)Experiments (Choi et al., 2007): Simulations (Im and Huang, 2009):
500025001250
Stress relaxation in both x and y directions from the edges.The length of edge relaxation g gdepends on the elastic modulus of the substrate.No wrinkles at the corners, parallel wrinkles along the edges andwrinkles along the edges, and transition to zigzag and labyrinth in large films.
Wrinkling of periodically patterned filmsE i t (B d t l 1998)Experiments (Bowden et al., 1998):
Simulations (Im and Huang, 2009)
Wrinkling of Single-Crystal Thin Films30 nm SiGe film on BPSG at 750°C30 nm SiGe film on BPSG at 750 C(Hobart et al., 2000; Yin et al., 2002 & 2003;R.L Peterson, PhD thesis, Princeton, 2006)
100 nm Si on PDMS (Song et al., 2008)100 nm Si on PDMS (Song et al., 2008)
(100)(010)
Spectra of initial growth rate
σ2[010] θ =0: θ = 45°:
σ1[100]
Im and Huang, JMPS 2008.
Energy Spectra of Equilibrium Wrinkles
σ2[010] θ =0: θ = 45°:
σ1[100]
Evolution of orthogonal wrinkle patterns003.021 −==σσ 003.021 σσ
t = 0:RMS = 0.0057λave= 38.8
t = 105 :RMS = 0.0165λave= 44.8
t = 5 × 105 :RMS = 0.4185λave= 47.1
t 106 t = 107 : t = 108 :t = 106 :RMS = 0.4676λave= 50.8
t = 107 :RMS = 0.5876λave= 56.4
t = 108 :RMS = 0.5918λave= 56.6
Im and Huang, JMPS 2008.
Effect of stress magnitude
Im and Huang, JMPS 2008.
Checkerboard wrinkle pattern
0
0.5
1
0
20
40
0
20
40
−1
0.5
4040
00178.021 −==σσ RMS = 0.05286 and λave= 55.6
Transition of local buckling mode from spherical to cylindrical as the compressive stress/strain increases and/or as the wrinkle amplitude increases.
Im and Huang, JMPS 2008.
Transition of wrinkle patterns
Im and Huang, JMPS 2008.
Summary• Viscoelastic wrinkling
– Three-stage evolutionThree stage evolution– Power-law coarsening
Near equilibrium patterns– Near-equilibrium patterns
Di i kl tt• Diverse wrinkle patterns– Anisotropic patterns under biaxial stresses– Nonuniform patterns– Wrinkling of anisotropic crystal thin films